112,586 research outputs found

    Near-Linear Time Insertion-Deletion Codes and (1+ε\varepsilon)-Approximating Edit Distance via Indexing

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    We introduce fast-decodable indexing schemes for edit distance which can be used to speed up edit distance computations to near-linear time if one of the strings is indexed by an indexing string II. In particular, for every length nn and every ε>0\varepsilon >0, one can in near linear time construct a string IΣnI \in \Sigma'^n with Σ=Oε(1)|\Sigma'| = O_{\varepsilon}(1), such that, indexing any string SΣnS \in \Sigma^n, symbol-by-symbol, with II results in a string SΣnS' \in \Sigma''^n where Σ=Σ×Σ\Sigma'' = \Sigma \times \Sigma' for which edit distance computations are easy, i.e., one can compute a (1+ε)(1+\varepsilon)-approximation of the edit distance between SS' and any other string in O(npoly(logn))O(n \text{poly}(\log n)) time. Our indexing schemes can be used to improve the decoding complexity of state-of-the-art error correcting codes for insertions and deletions. In particular, they lead to near-linear time decoding algorithms for the insertion-deletion codes of [Haeupler, Shahrasbi; STOC `17] and faster decoding algorithms for list-decodable insertion-deletion codes of [Haeupler, Shahrasbi, Sudan; ICALP `18]. Interestingly, the latter codes are a crucial ingredient in the construction of fast-decodable indexing schemes

    On the lengths of divisible codes

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    In this article, the effective lengths of all qrq^r-divisible linear codes over Fq\mathbb{F}_q with a non-negative integer rr are determined. For that purpose, the Sq(r)S_q(r)-adic expansion of an integer nn is introduced. It is shown that there exists a qrq^r-divisible Fq\mathbb{F}_q-linear code of effective length nn if and only if the leading coefficient of the Sq(r)S_q(r)-adic expansion of nn is non-negative. Furthermore, the maximum weight of a qrq^r-divisible code of effective length nn is at most σqr\sigma q^r, where σ\sigma denotes the cross-sum of the Sq(r)S_q(r)-adic expansion of nn. This result has applications in Galois geometries. A recent theorem of N{\u{a}}stase and Sissokho on the maximum size of a partial spread follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.Comment: 17 pages, typos corrected; the paper was originally named "An improvement of the Johnson bound for subspace codes

    On the equivalence of linear sets

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    Let LL be a linear set of pseudoregulus type in a line \ell in Σ=PG(t1,qt)\Sigma^*=\mathrm{PG}(t-1,q^t), t=5t=5 or t>6t>6. We provide examples of qq-order canonical subgeometries Σ1,Σ2Σ\Sigma_1,\, \Sigma_2 \subset \Sigma^* such that there is a (t3)(t-3)-space ΓΣ(Σ1Σ2)\Gamma \subset \Sigma^*\setminus (\Sigma_1 \cup \Sigma_2 \cup \ell) with the property that for i=1,2i=1,2, LL is the projection of Σi\Sigma_i from center Γ\Gamma and there exists no collineation ϕ\phi of Σ\Sigma^* such that Γϕ=Γ\Gamma^{\phi}=\Gamma and Σ1ϕ=Σ2\Sigma_1^{\phi}=\Sigma_2. Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.Comment: Preprint version. Referees' suggestions and corrections implemented. The final version is to appear in Designs, Codes and Cryptograph

    On effective sigma-boundedness and sigma-compactness

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    We prove several theorems on sigma-bounded and sigma-compact pointsets. We start with a known theorem by Kechris, saying that any lightface \Sigma^1_1 set of the Baire space either is effectively sigma-bounded (that is, covered by a countable union of compact lightface \Delta^1_1 sets), or contains a superperfect subset (and then the set is not sigma-bounded, of course). We add different generalizations of this result, in particular, 1) such that the boundedness property involved includes covering by compact sets and equivalence classes of a given finite collection of lightface \Delta^1_1 equivalence relations, 2) generalizations to lightface \Sigma^1_2 sets, 3) generalizations true in the Solovay model. As for effective sigma-compactness, we start with a theorem by Louveau, saying that any lightface \Delta^1_1 set of the Baire space either is effectively sigma-compact (that is, is equal to a countable union of compact lightface \Delta^1_1 sets), or it contains a relatively closed superperfect subset. Then we prove a generalization of this result to lightface \Sigma^1_1 sets.Comment: arXiv admin note: substantial text overlap with arXiv:1103.106
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